﻿the Hatchet Planimeter. 2(57 



Also from (1) by differentiating, 



d% _dr 

 sin# e' 



Now if yjr be the angle between OP and the tangent PQ, 

 the inclinations of the rod to PQ are yjr — + <$>' at P, and 

 \}r — -{- ft + dcf) 1 at Q, and the length P Q is ds; 



therefore d<fi _ ds 



sin (■f—0 + <p')~ e' 

 or 



cd(j>'= {sin yfr cos (0 — ft) — cos yjr sin (0 — <£')}<£? 



= rd0 cos (#-<£') +dr sin (0-c//). 



Substituting this value of c/<£' in (3), 



cd6 -rdd cos {0-$') 7 , , c(d0-d<f>) 



sin (0-f) sin(<9-<£) , 



or 



~^—Ta — XT=^r( 1— -)cot — r- + -r( 1+- ) tan -V 1 

 sm(0 — (f>) 2\c/ 2 2 \ cj 2 



d0(. r\ r 0-$ d0/. r\ ' 0-<f> 

 = Y{ 1 -V eCC ° t 2 + -2\} + c) e cttm -2^ 

 by (2); 



/. d0-d$= [(l^rpcos 2 ^ + (l + ^-Srin-^JiW 



-i[(i-9- £ +(i+^> 



-(a4 + 3-t + ---) cos( ^^- 



Oi 



r' 2 d0 r 4 r 6 



+ («; + 85? + •••)"<»<'-*>■ • • ( 4 ) 



This equation gives the change in the direction of the rod 

 when the tracing-point has moved round an elementary 



T2 



